It seems like the verb that uses「~て」should be affected by an adverb or an auxiliary verb unless we have a string of「~て、~て ...」。Grammatically, for an adjective to affect a verb should require abstracting the action to a concept and treating the abstracted concept as an object (i.e.: 「~ることがほしい」、but that would be wrong and awkward also). Obviously,「~たい」could also be used, but that isn't the issue. Right? What I am trying to figure out is how this functions on a grammatical level. As I understand it, we have a continuative verb form（「~て」）immediately followed by an adjective. With the「~て欲しいです」construction, we then have what converts into English grammar as, for example, "run and is wanted" which makes no sense. Trying to process this using EN and JP, it doesn't make any sense to me either way. Is this just some exception to the rest of the grammatical system as a fixed construction? What is the grammatical explanation for how this can be correct?
@A.Ellett's explanation is correct. There are various auxiliaries that are grammatical for specific verb forms. There's a small selection of examples for the て形 on Wikipedia, and there are some other auxiliary examples that are specifically compatible with the 連用形 ("masu stem"), also on Wikipedia.
I'm not sure if it's a comprehensive list, but the DBJG lists many auxiliaries for the て形 (p. 593, Appendix 4: "F. Vte+__") which you can learn more about from.
EDIT: This answer is still correct, but I now understand the author's confusion. Let this be clear: 欲しい (adjective) ≠ 〜て欲しい (auxiliary verb). There is no grammatical syntax pattern that allows arbitrary i-adjectives to append the "te-form". Rather, there is a small list of auxiliary verbs (less than 20) that exist for attaching to the te-form. Somewhere in the history of the language, 欲しい somehow became one of the te-form auxiliary verbs; but the reason for this has nothing to do with the word being an adjective.
tl;dr: You cannot randomly attach i-adjectives to the te-form of a verb. You can only attach one of the auxiliary verbs intended for the te-form, and I assume there are less than 20 such verbs, one being 〜て欲しい.
Citation: Makino, Seiichi; Tsutsui, Michio (1989). A Dictionary of Basic Japanese Grammar (80 ed.). Tokyo, Japan: The Japan Times., p. 593, ISBN 978-47-89004-54-1.
It seems you're thinking about the て-form a bit too narrowly. It doesn't just map onto one grammatical structure in English.
The ~てほしい construction works just as the ~てください construction does.
I want you to wait.
They're both expressing that you want/desire something from someone else. One is a polite request; the other expresses your desire, but isn't necessarily a request for anything.
I'm not sure if I'm reading what you're saying correctly, but it seems you might be thinking that 待ってほしい and 待ちたい mean the same thing. This isn't at all the case. (And, I apologize if I've misunderstood what you were trying to say on this point.) Again, 待ってほしい means "I want you to wait". 待ちたい means "I want to wait."
Don't just think of the て-form as expressing "[doing something] and". That's a too narrow reading. Instead, the meaning of the て-form can't really be resolved until you read/hear what follows.
~ることがほしい doesn't really make sense. When you use the structure ~がほしい the ~ isn't some abstraction; it's something concrete: りんごがほしい.